On the existence of recursive utility

Recursive utility functions are often defined implicitly, as the solution of a functional equation known as the Koopmans recursion. As conditions for a unique solution can be demanding, a recent literature has tackled the problem of multiplicity, advocating that one focus attention on either the least or greatest solution, which possess a number of attractive properties. While a least solution can be shown to exist, I show that in the most general formulation of the problem, a greatest solution need not. The goal of the paper is to provide a sufficient condition for the existence of a greatest solution, which I term the weakly contractive property and which generalizes the usual contractive property sufficient for uniqueness.